Classification from a Computable Viewpoint

TL;DR

This paper reviews recent developments in the classification of mathematical structures within computable structure theory, highlighting which classes can or cannot be classified using computable invariants.

Contribution

It provides an overview of recent results on the limits and possibilities of classifying structures in the computable setting.

Findings

Certain classes admit computable classification

Some classes resist classification by computable invariants

The work bridges model theory and computability theory

Abstract

Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence in terms of relatively simple invariants. Where this is impossible, it is useful to have concrete results saying so. In model theory and descriptive set theory, there is a large body of work, showing that certain classes of mathematical structures admit classification, while others do not. In the present paper, we describe some recent work on classification in computable structure theory.